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ANALYSIS ON THE BLOW-UP OF SOLUTIONS TO A CLASS OF

INTEGRABLE PEAKON EQUATIONS

ROBIN MING CHEN, FEI GUO, YUE LIU, AND CHANGZHENG QU

Abstract. We investigate the blow-up mechanism of solutions to a class of quasilinear

integrable equations which could possess peakons. The dynamics of the blow-up quantity

along the characteristics is established by the Riccati-type differential inequality which

involves the interaction among three parts: a local nonlinearity, a nonlocal term, and a

term stemming from the weak linear dispersion. To analyse the interplay among these

quantities, we provide two different approaches. The first one is designed for the case

when the equations do not exhibit a weak linear dispersion and hence focuses on the

interplay between the first two parts. The method is based on a refined analysis on

either evolution of the solution u and its gradient ux, that is, Cu ± ux or the growth

rate of the relative ratio ux/u. The second one handles the general situation when all

of three parts are present. The idea is to extract the “truly” blow-up component from

the Riccati-type differential inequality and utilizes the Morawetz-type identity or higher

order conservation laws to show that such a component blows up in finite time before

the other component degenerates.

Keywords: Degasperis-Procesi equation, Novikov equation, generalized modified Camassa-

Holm equation, blow-up, peakon, wave-breaking.

AMS Subject Classification (2010): 35G25, 35B44, 35Q35

1. Introduction

Wave motion can be distinguished in two main classes: the hyperbolic waves and dis-

persive waves [30]. One of the common characteristics that both these wave models exhibit

is the (finite-time) blow-up: there is a time T <∞ such that the certain norm of solution

of the model equations becomes unbounded as t ↑ T . Some prototypes include the shock

waves which are propagating discontinuities in the dependent variables – a nonlinear fea-

ture of the hyperbolic waves – which is caused by progressively nonlinear steepening of

the wave profiles [14]; and the dispersive blow-up for dispersive waves, which is a focusing

type of behavior that is due to propensity of the dispersion relation so that infinitely many,

widely spaced small disturbances may coalesce locally in space-time [2].

On the one hand, dispersion is known to spread out waves and make them decay in time,

delaying the onset of blow-up. In fact if the dispersion can be strong enough to overcome

the nonlinear effects so that wave interaction takes place at a fast rate over a short time,

then the smoothness of the waves can persist, excluding the possibility of blow-ups. One

of the best known examples may be found in the context of water waves, namely the

celebrated Korteweg-de Vries (KdV) equation [21]. On the other hand, it is observed that

there is another effect, namely the nonlocal (smoothing) effect, which can help maintain

Date: February 8, 2015.

1

2 R.M. CHEN, F. GUO, Y. LIU, AND C.Z. QU

the regularity while waves propagate and hence prevent them from blowing up, even when

dispersion is weak or absent. See, for example, the Benjamin-Bona-Mahoney (BBM)

equation [1]. As the nonlinearity becomes stronger and dominates over the dispersion

and nonlocal effects singularities may occur in the sense of wave-breaking, i.e., the wave

profile remains bounded, but its slope becomes unbounded. Examples can be found in the

Whitham equation [9, 30], Camassa-Holm (CH) equation [5, 13], the Degasperis-Procesi

(DP) equation[16, 13], and the Novikov equation [24], etc. When the dispersion is weak

and the nonlinearity reaches a certain balance with the nonlocal terms, the curvature (the

second derivative of the solution) may blow up in finite time. See, for instance, the modified

Camassa-Holm (mCH) equation [25]. Understanding the wave-breaking mechanism such

as when a singularity can form and what the nature of it is not only presents fundamental

importance from mathematical point of view but also is of great physical interest, since

it would help provide a key-mechanism for localizing energy in conservative systems by

forming one or several small-scale spots. For instance, in fluid dynamics, the possible

phenomenon of finite time breakdown for the incompressible Euler equations signifies the

onset of turbulence in high Reynolds number flows.

The purpose of this paper is to study finite-time blow-up of solutions for a class of

quasilinear dispersive equations. These equations include the DP equation, the Novikov

equation, and the generalized modified Camass-Holm (gmCH) equation, all of which ex-

hibit nonlocal nonlinearities and nonlinear dispersion. A distinctive feature, which is also

a primary reason for the interest in these equations is that they are integrable models

for the breakdown of regularity. Moreover, these equations admit a remarkable variety of

the so-called “peakon” solutions – peaked traveling wave solutions with a discontinuous

derivative at crest [15, 20, 26]. Physically, due to the relevance of many of the preceding

equations to water waves, those peakons reveal some similarity to the well-known Stokes

waves of greatest height – the traveling waves of maximum possible amplitude that are

solutions to the governing equations for irrotational water waves [7, 29].

We would first like to review some basic integrability properties of the three equations

mentioned above. The well-studied DP equation

mt + umx + 3uxm = 0, m = u− uxx (1.1)

is completely integrable with the associated Lax pair and admits a bi-Hamiltonian struc-

ture [15, 16]

mt = B0δH−1

δm= B1

δH0

δm,

where

B0 = −1

6∂x(1− ∂2

x)(4− ∂2x), B1 = −9

2m2/3∂xm

1/3(∂x − ∂3x)−1m1/3∂xm

2/3,

H−1 =

∫Ru3 dx, H0 =

∫Rm dx.

Furthermore, the DP equation admits the following conserved density [23]

H1[u] =

∫Ry(t, x)v(t, x)dx =

∫Ry0(x)v0(x)dx, (1.2)

where y = (1− ∂2x)u, v = (4− ∂2

x)−1u.

BLOW-UP TO INTEGRABLE PEAKON EQUATIONS 3

In contrast to the DP equation, the Novikov equation [24] exhibits a cubic nonlinearity

mt + u2mx + 3uuxm+ γux = 0, m = u− uxx. (1.3)

It can also be written in a bi-Hamiltonian form [20]

mt = J0δH1

δm= J1

δH0

δm,

where

J0 =1

4(1−D2

x)m−1Dxm−1(1−D2

x),

J1 = −1

3(3mDx + 2mx)(4Dx −D3

x)−1(3mDx +mx) +1

3γDx,

and

H1 =

∫Rumdx, H0 =

∫R

(umD−1

x (mut)−1

2γumD−1

x (m(1−D2x)−1ux)

)dx.

The gmCH equation [18]

mt + k1

((u2 − u2

x)m)x

+ k2(2uxm+ umx) + γux = 0, m = u− uxx (1.4)

involves both quadratic and cubic nonlinearities, and is obtained by applying tri-Hamiltonian

duality to the bi-Hamiltonian Gardner equation. Note that equation (1.4) reduces to the

CH equation when k1 = 0, k2 = 1, and to the mCH equation when k1 = 1, k2 = 0, respec-

tively. The gmCH equation (1.4) also admits the Lax pair and has the bi-Hamiltonian

form [25, 26]

mt = JδH1

δm= K

δH2

δm,

where

J = −k1∂xm∂−1x m∂x −

1

2k2(m∂x + ∂xm)− 1

2γ∂x, and K =

1

4

(∂3x − ∂x

)H1 =

∫Rumdx, and H2 = k1I1 + 2k2I2 + 2γ

∫Ru2dx (1.5)

with

I1 =

∫R

(u4 + 2u2u2

x −1

3u4x

)dx, I2 =

∫R

(u3 + uu2

x

)dx.

Unlike the semilinear dispersive systems, for instance, the KdV and Schrodinger equa-

tions, where in many cases the linear dispersion dominates over the nonlinearity and a

contraction principle can be applied even in a very low-regularity regime to obtain well-

posedness, and consequently the smoothness of the solution propagates with the help of

conservation laws, the equations considered here are all quasilinear with weak linear dis-

persion, suggesting that well-posedness can only be established in a high regularity regime,

and the initial profile can determine the existence time and the regularity of the solution

map rather strongly.

There has been many of study of the finite-time blow-up of equations (1.1), (1.3), (1.4)

and some related models. We do not attempt to exhaust all the literatures. One can

refer to [6, 8, 9, 10, 11, 12, 16, 19, 22, 28] and the references therein for details. The

main idea used in the analysis is to trace the dynamics of the blow-up quantity along

the characteristics. Due to the connection between u and the momentum density m,


these equations inherit a nonlocal structure and can be reformulated in a weak form

of nonlinear nonlocal transport type. From the transport theory, the blow-up criteria

assert that singularities are caused by the focusing of characteristics, which involve the

information on the gradient ux and m. Roughly speaking, the dynamics of the blow-up

quantity B = B(u, ux,m) along the characteristics is governed by an equation

B′(t) . −B2 + f(u,m)(C2u2 − u2

x

)+N (u, ux,m) +Dγ , (1.6)

where N (u, ux,m) is the nonlocal part, which usually consists of convolutions against

the kernel p(x) = 12e−|x|, the fundamental solution of (1 − ∂2

x)−1 on R, and Dγ comes

from the weak linear dispersion and is of lower order. In many classical cases there is

no linear dispersion, and hence Dγ = 0. Standard approaches seek certain conservation

laws (like, for instance, the conservation of the H1-norm of u, the persistence of the sign

of m, antisymmetry, etc.) to control the local quantities involving u,m as well as the

nonlocal term N by constants. In particular, for the CH and DP equations, the term

f(u,m)(C2u2 − u2

x

)can be replaced by a function f(u) depending solely on u, and hence

the dynamics of B follows a Riccati-type inequality B′ . −B2 + C, which leads to a

finite-time blow-up provided B is sufficiently negative initially. As such approaches make

an intensive use of the “global” information of solutions, the blow-up mechanism ignores

the local structure of the solutions.

Recently Brandolese and Cortez [3, 4] introduced a new type of blow-up criteria in

the study of the CH-type equations which highlights how local structure of the solution

affects the blow-ups. Their argument relies heavily on the fact that the convolution terms

are quadratic and positively definite, and that the convolution kernel p(x) satisfies that

p ± κpx ≥ 0 for |κ| ≤ 1. For the DP equation considered here, however, one needs to

deal with convolutions against p±√

32px, and more seriously, for the Novikov and gmCH

equations, the convolution contains cubic terms which do not have a lower bound in terms

of the local terms. For this reason, it is not clear whether a purely local condition on the

initial data can generate finite-time blow-ups.

We present two different ideas to investigate the breakdown mechanism of equations

(1.1), (1.3) and (1.4). The first approach deals particularly with the dispersionless situation

γ = 0. We look for some global property, namely the sign persistence of momentum density

m, to bound the nonlocal term N . Thus from (1.6), the blow-up can be deduced by the

interplay between u and ux. More precisely, this motivates us to carry out a refined

analysis of the characteristic dynamics of M = Cu − ux and N = Cu + ux. For the DP

and Novikov equations, the estimates of M and N can be closed in the form of

M ′(t) ≥ −g(u)MN +N1, N ′(t) ≤ g(u)MN +N2, (1.7)

where g(u) ≥ 0 and the nonlocal terms Ni (i = 1, 2) can be bounded in terms of certain

higher order conservation laws. From this the monotonicity ofM andN can be established,

and hence the finite-time blow-up follows. The situation for the gmCH equation is more

delicate. The estimates (1.7) are not available for M and N . But note that an alternative

way to show that C2u2−u2x ≤ 0 is to track the relative ratio |ux/u|, and to prove that the

ratio stays sufficiently large. Physically this amounts to considering the local oscillation of

solutions. Intuitively, one would expect that fast oscillation causes breakdown of solutions.


It turns out that the dynamics of ux/u can be put in a rather clean form. However the

inhomogeneity of the nonlinearities in the equation makes it difficult to extract a clear

ratio condition out of the estimate. By performing a vertical shift of the solution, we

are able to make the estimates homogeneous, which in turn provides the desired ratio

property. The main theorems along this line are Theorem 2.1, Theorem 3.1 and Theorem

4.1.

The other approach we adopt has some similarity to the classical one and can be ap-

plied to the more general situations with weak linear dispersion. We will focus on the

Novikov and the gmCH equations since this type of wave-breaking for the DP equation

has been addressed in [23]. There are two sources of difficulties in this approach. Firstly,

the convolution part in N contains cubic nonlinearities in ux, and thus it requires some

higher-order conservation laws. Fortunately, for the dispersionless Novikov and the gmCH

equations, there exist some conserved quantities that bound the L4 norm of ux (cf. (3.4)

for Novikov and (1.5) for gmCH). This together with the H1 estimates of u controls the

cubic terms in the convolution. For the general Novikov equation with linear dispersion,

the previous functional in the dispersionless case is not conserved. Instead, we can still

control the term ‖ux‖4L4 by a Morawetz-type identity derived for a modified functional

I(t) in (3.13). Since the evolution of I is bounded by ‖ux‖2L4 , that is

d

dt‖ux(t)‖4L4 . ‖ux(t)‖2L4 ,

this implies that ‖ux‖L4 is bounded by√t. In this way the cubic convolution can be

bounded accordingly.

The second difficulty, which is more serious, lies in the fact that the needed local es-

timates in (1.6) involve the L∞ control of ux or m, which can not be inferred from the

conservation laws. Our idea is to avoid examining the dynamic of B by extracting the

“truly” blow-up component from it and to look at the dynamics of that component in-

stead. More precisely, for the Novikov equation, the blow-up quantity is B = uux. The

H1 conservation indicates that u remains bounded for all time. Hence for B to blow up,

it suffices to show that ux blows up while u does not degenerate to zero in finite time.

It turns out that the dynamics of u and ux are much simpler, cf. (3.6), and the local

estimates do not require an L∞ bound on ux. Therefore we are able to push ux to infinity

before u shrinks down to zero. The corresponding result is stated in Theorem 3.2.

On the other hand, the blow-up quantity for the gmCH equation is B = (k1m+ k2)ux,

in which both m and ux can potentially blow up in finite time. By checking their dynamics

individually we find that the dynamic equation for m has a very simple structure m′(t) =

q(m)ux, where q(m) is a quadratic polynomial in m. Moreover, although the equation for

u′x still contains local terms involving ux, it can be made of a definite sign with the help of

the conservation laws. This way ux will be monotone. With appropriate choice of initial

data, the later dynamics of m satisfies m′(t) & q(m). Solving this differential inequality

one obtains a finite-time blow-up of m. Due to the monotonicity of ux and the proper

choice of its initial value, ux can be made uniformly away from zero. Hence B blows up.

The details can be found in Theorem 4.2.


The rest of the paper is organized as follows. Section 2 deals with the DP equation and

formulates the wave-breaking mechanism in Theorem 2.1. Sections 3 and Section 4 are

devoted to the blow-up for the Novikov and gmCH equations respectively, with an em-

phasis on the role of the weak linear dispersion. The blow-up results in the absence of the

weak dispersion are illustrated in Theorem 3.1 for the Novikov equation and Theorem 4.1

for the gmCH equation, respectively. In the general case when the weak linear dispersion

is at present, the breakdown mechanisms are set up in Theorem 3.2 and Theorem 4.2.

Notation. In the sequel, we denote by ∗ the convolution. For 1 ≤ p < ∞, the norms

in the Lebesgue space Lp(R) is ‖f‖p =( ∫

R |f(x)|pdx) 1

p, the space L∞(R) consists of all

essentially bounded, Lebesgue measurable functions f equipped with the norm ‖f‖∞ =

infµ(e)=0

supx∈R\e

|f(x)|. For a function f in the classical Sobolev spaces Hs(R) (s ≥ 0) the norm

is denoted by ‖f‖Hs . We denote p(x) = 12e−|x| the fundamental solution of 1 − ∂2

x on R,

and define the two convolution operators p+, p− as

p+ ∗ f(x) =e−x

2

∫ x

−∞eyf(y)dy

p− ∗ f(x) =ex

2

∫ ∞x

e−yf(y)dy.

(1.8)

Then we have the relations p = p+ + p−, px = p− − p+.

2. Wave-breaking for the Degasperis-Procesi equation

We begin with the DP equation (1.1). The characteristics q(t, x) associated to the DP

equation is governed by{qt(t, x) = u(t, q(t, x)),

q(0, x) = x,x ∈ R, t ∈ [0, T ). (2.1)

One can easily check that q ∈ C1([0, T )×R,R) with qx(t, x) > 0 for all (t, x) ∈ [0, T )×R.

Furthermore the potential m = u− uxx satisfies

m(t, q(t, x))q3x(t, x) = m0(x), (t, x) ∈ [0, T )× R, (2.2)

which implies that the zeros and the sign of m are preserved under the flow.

The precise blow-up criterion for the DP equation can be formulated as

Lemma 2.1. [23] Let u0 ∈ Hs(R) for s > 3/2. The blow-up of solution in finite time

T ∗ < +∞ occurs if and only if

limt→T ∗

infx∈R

ux(t, x) = −∞. (2.3)

It is easy to check that the derivatives of u and ux along the characteristics can be

obtained from the following computations

ut + uux = −px ∗(

3

2u2

), (2.4)

uxt + uuxx =3

2u2 − u2

x − p ∗(

3

2u2

). (2.5)


For wave-breaking of the DP equation, one would like to know under what conditions

on the initial data u0(x), ux approaches −∞ in finite time. From equation (2.5), it suffices

to show that ux can have super-linear decay rate along certain characteristics. Our goal is

to find some initial data so that at later time along the corresponding characteristics u2x

outgrows 32u

2. This can be done by tracking the dynamics of the two quantities√

32u±ux.

Notice that the dynamic equation (2.4) consists only of the convolution term, which can

be controlled by using the Young inequality and the uniform L2 norm in the following

p± ∗(

3

2u2

)≤ ‖p±‖∞

∥∥∥∥3

2u2

∥∥∥∥1

=3

4‖u‖22 ≤ 3H1[u0], (2.6)

where H1[u0] is given in (1.2) and we have used the following estimate (see [23])

1

4‖u(t)‖22 =

1

4

∫Ru2(ξ)dξ ≤

∫R

1 + ξ2

4 + ξ2u2(ξ)dξ =

∫Ryvdx = H1[u0].

The wave-breaking result is now formulated as follows.

Theorem 2.1. Let u0 ∈ Hs(R) for s > 3/2. Suppose that there exists a point x1 ∈ Rsuch that

u0,x(x1) < −√

3

2|u0(x1)| − C, (2.7)

where

C =

√√√√3

(√3

2− 1

)H1[u0]. (2.8)

Then the corresponding solution u(t, x) blows up in finite time with an estimate of the

blow-up time T ∗ as

T ∗ ≤ 1

2Clog

√u2

0,x(x1)− 32u

20(x1) + C√

u20,x(x1)− 3

2u20(x1)− C

.

Proof. We track the dynamics ofM1(t) =

(√3

2u− ux

)(t, q(t, x1)) andN1(t) =

(√3

2u+ ux

)(t, q(t, x1)) along the characteristics and apply the convolution estimate (2.6) to obtain

M ′1(t) = −M1N1 +

(√3

2+ 1

)p+ ∗

(3

2u2

)−

(√3

2− 1

)p− ∗

(3

2u2

)

≥ −M1N1 −

(√3

2− 1

)p− ∗

(3

2u2

)

≥ −M1N1 − 3

(√3

2− 1

)H1[u0] = −M1N1 − C2,

N ′1(t) = M1N1 +

(√3

2− 1

)p+ ∗

(3

2u2

)−

(√3

2+ 1

)p− ∗

(3

2u2

)

≤M1N1 + 3

(√3

2− 1

)H1[u0] = M1N1 + C2,


where C is defined in (2.8), and ′ denotes the derivative ∂t + u∂x. Then we have

M ′1(t) ≥ −M1N1 − C2, N ′1(t) ≤M1N1 + C2.

The expected monotonicity conditions on M and N indicate that we would like to have

M1N1(t) + C2 < 0.

Therefore it is found from (2.7) that the initial data satisfies

3

2u2

0(x1)− u20,x(x1) < −C2,

√3

2u0(x1) + u0,x(x1) < 0, (2.9)

In this way, we know that along the characteristics emanating from x1,

M1(0) > 0, N1(0) < 0, M ′1(0) > 0, N ′1(0) < 0.

Therefore over the time of existence it always holds that

M ′1(t) > 0, N ′1(t) < 0, M1N1(t) < −C2.

Hence we may consider the evolution of the quantity h1(t) =√−M1N1(t). By using the

estimateM1 −N1

2≥ h1, we have

h′1 = −M′1N1 +M1N

′1

2√−M1N1

≥ −(M1N1 + C2)(M1 −N1)

2√−M1N1

= (h21 − C2)

(M1 −N1)

2√−M1N1

≥ h21 − C2,

which implies that h1(t)→ +∞ as t→ T ∗ with estimate on T ∗ given by

T ∗ ≤ 1

2Clog

h1(0) + C

h1(0)− C.

Note that h1(t) ≤ −ux(t, q(t, x1)). Hence h1(t) → ∞ as t → T ∗ implies the finite time

blow up ux(t, q(t, x1))→ −∞ as t→ T ∗. �

3. Wave-breaking for the Novikov equation

The Novikov equation (1.3) can also be written as

ut − utxx + 4u2ux = 3uuxuxx + u2uxxx − γux. (3.1)

The associated characteristics is{qt(t, x) = u2(t, q(t, x)),

q(0, x) = x,x ∈ R, t ∈ [0, T ). (3.2)

If γ = 0, then the dynamics of the momentum density is

m(t, q(t, x))q3/2x (t, x) = m0(x), (t, x) ∈ [0, T )× R. (3.3)

In this case, equation (3.1) admits the following two conserved densities, which will be

important in our blow-up analysis.

H1[u(t)] =

∫R

(u2 + u2x) dx = H1[u0],

H2[u(t)] =

∫R

(u4 + 2u2u2

x −1

3u4x

)dx = H2[u0].

(3.4)


The blow-up criterion for the Novikov equation is formulated as follows.

Lemma 3.1. [28] Let u0 ∈ Hs(R) for s > 3/2. The solution of equation (3.1) with initial

data u0 blows up in finite time T ∗ if and only if

limt→T ∗

infx∈R{u(t, x)ux(t, x)} = −∞. (3.5)

3.1. Dynamics along the characteristics. Let us compute the dynamics of a few im-

portant quantities along the characteristics q(t, x). Denote ′ to be the derivative ∂t+u2∂xalong the characteristics.

Lemma 3.2. Let u0 ∈ Hs(R), s ≥ 3. Then u(t, q(t, x)), ux(t, q(t, x)) and (uux)(t, q(t, x))

satisfy the following integro-differential equations

u′(t) = −γp ∗ ux +1

2

[p+ ∗ (u− ux)3 − p− ∗ (u+ ux)3

],

u′x(t) =u

2(u2 − u2

x) + γ (u− p ∗ u)

− 1

2

[p+ ∗ (u− ux)3 + p− ∗ (u+ ux)3

],

(uux)′(t) =u2

2(u2 − u2

x) + γu2 − γ [ux(p ∗ ux) + u(p ∗ u)]

− 1

2

[(u− ux)p+ ∗ (u− ux)3 + (u+ ux)p− ∗ (u+ ux)3

].

(3.6)

Proof. The first one we look at is

u′(t) = −p ∗ (3uuxuxx + 2u3x + 3u2ux + γux)

= −px ∗(

3

2uu2

x + u3

)− 1

2p ∗ u3

x − γp ∗ ux

= p+ ∗(

3

2uu2

x + u3 − 1

2u3x

)− p− ∗

(3

2uu2

x + u3 +1

2u3x

)− γp ∗ ux.

The first term can be calculated in the following.

p+ ∗(

3

2uu2

x + u3 − 1

2u3x

)=

1

2p+ ∗ (u− ux)3 +

1

2p+ ∗

(u3 + 3u2ux

)=

1

2p+ ∗

((u− ux)3 + u3

)+

1

4u3 − 1

2p+ ∗ u3

=1

2p+ ∗ (u− ux)3 +

1

4u3.

Similarly, the second term is

p− ∗(

3

2uu2

x + u3 +1

2u3x

)=

1

2p− ∗ (u+ ux)3 +

1

4u3.

Putting together, we obtain

u′(t) = −γp ∗ ux +1

2

(p+ ∗ (u− ux)3 − p− ∗ (u+ ux)3

). (3.7)

Next we estimate u′x.

u′x(t) = −1

2uu2

x + u3 − p ∗(

3

2uu2

x + u3

)− 1

2px ∗ u3

x − γpx ∗ ux


= −1

2uu2

x + u3 + γ(u− p ∗ u)− p+ ∗(

3

2uu2

x + u3 − 1

2u3x

)− p− ∗

(3

2uu2

x + u3 +1

2u3x

).

The above computation then leads to

u′x(t) =u

2(u2 − u2

x) + γ(u− p ∗ u)− 1

2

[p+ ∗ (u− ux)3 + p− ∗ (u+ ux)3

]. (3.8)

Finally we turn our attention to the blow-up quantity uux.

(uux)′(t) = u2

(u2 − 1

2u2x

)+ γu2 − γ[ux(p ∗ ux) + u(p ∗ u)]

− (u− ux)p+ ∗(

3

2uu2

x + u3 − 1

2u3x

)− (u+ ux)p− ∗

(3

2uu2

x + u3 +1

2u3x

).

Hence we obtain

(uux)′(t) =u2

2(u2 − u2

x) + γu2 − γ [ux(p ∗ ux) + u(p ∗ u)]

− 1

2

[(u− ux)p+ ∗ (u− ux)3 + (u+ ux)p− ∗ (u+ ux)3

].

(3.9)

�

3.2. Wave-breaking data. In this section we will consider two different classes of initial

data related to the appearance of the weak linear dispersion, and establish the needed

estimates for the convolution terms. The following result for ODE theory will be useful

in our proof of blow-up.

Lemma 3.3. Let f ∈ C1(R), a > 0, b > 0 and f(0) >√

ba . If f ′(t) ≥ af2(t)− b, then

f(t)→ +∞ as t→ t∗ ≤ 1

2√ab

log

f(0) +√

ba

f(0)−√

ba

.

3.2.1. Sign-changing momentum. First we consider the case when there is no weak linear

dispersion and so the momentum density preserves its sign along the characteristics. We

choose the initial momentum density m0 that changes sign at exactly one point.

Theorem 3.1. Let γ = 0 and m0 ∈ Hs(R) for s > 1/2. Assume that there exists some

point x2 ∈ R such that u0(x2) > 0 and

m0(x) > 0, x < x2; m0(x2) = 0; m0(x) < 0, x > x2.

Moreoveru0(x2)

2

[u2

0(x2)− u20,x(x2)

]+K1 < 0,

where

K1 = 2

√H31 [u0]

2+

√3H1[u0]

(H2

1 [u0]− H2[u0]) . (3.10)




T ∗ ≤ 1√2K1u0(x2)

log

√u2

0,x(x2)− u20(x2) +

√2K1/u0(x2)√

u20,x(x2)− u2

0(x2)−√

2K1/u0(x2)

.

Proof. From the evolution equation (3.3) we know that along the characteristics q :=

q(t, x2) emanating from x2 we have

m(t, q) = 0.

Moreover

m(t, x) > 0, x < q(t, x2); m(t, x) < 0, x > q(t, x2).

Using the identities

u+ ux = 2p− ∗m; u− ux = 2p+ ∗m,

we obtain

(u+ ux)(t, x) < 0, x ≥ q(t, x2); (u− ux)(t, x) > 0, x ≤ q(t, x2).

In view of the above sign conditions we infer that along the characteristics q(t, x2),(p+ ∗ (u− ux)3

)(t, q) > 0,

(p− ∗ (u+ ux)3

)(t, q) < 0.

Furthermore from (3.7) we conclude that

u(t, q) > 0, u′(t, q) > 0. (3.11)

Now we use the conservation laws (3.4) to derive the needed convolution estimates.

Note that from (3.4) it follows that

‖ux‖44 = 3

∫R

(u4 + 2u2u2

x

)dx− 3H2[u0]

≤ 3(‖u‖2∞‖u‖22 + 2‖u‖2∞‖ux‖22

)− 3H2[u0]

≤ 3(H2

1 [u0]− H2[u0]),

where we have used the estimate ‖u‖∞ ≤1√2‖u‖H1 . Hence we can further estimate the

two convolution terms∣∣p± ∗ (u∓ ux)3∣∣ ≤ ‖p±‖∞ ∥∥(u∓ ux)3

∥∥1≤ 1

2

∥∥(|u|+ |ux|)3∥∥

1

≤ 2(‖u‖33 + ‖ux‖33

)≤ 2

√H31 [u0]

2+

√3H1[u0]

(H2

1 [u0]− H2[u0])

=: K1.

(3.12)

Using the above convolution estimates, the finite time blow-up is argued as follows. Denote

M2(t) = (u− ux)(t, q), N2(t) = (u+ ux)(t, q),

u(t) = u(t, q), ux(t) = ux(t, q).


Then from the previous calculation we have

M ′2(t) = − u2

(u2 − ux2) +[p+ ∗ (u− ux)3

](t, q) > − u

2M2N2(t),

N ′2(t) =u

2(u2 − ux2)−

[p− ∗ (u+ ux)3

](t, q) ≤ u

2M2N2(t) +K1.

So using the monotonicity of u (c.f. (3.11)), if the initial data satisfies

1

2u0(x2)M2(0)N2(0) +K1 < 0,

then it implies that

M ′2(t) > 0, N ′2(t) < 0,u

2(M2N2)(t) +K1 < 0.

Let h2(t) =√−M2N2(t). It in turn follows that

h′2(t) = −M′2N2 +M2N

′2

2√−M2N2

≥ −(u2M2N2 +K1

)(M2 −N2)

2√−M2N2

=(u

2h2

2 −K1

) (M2 −N2)

2√−M2N2

≥ u

2h2

2 −K1 ≥u0(x2)

2h2

2 −K1.

It is deduced from Lemma 3.3 that h2(t)→ +∞ as t→ T ∗ with estimate on T ∗ given by

T ∗ ≤ 1√2K1u0(x2)

log

(h2(0) +

√2K1/u0(x2)

h2(0)−√

2K1/u0(x2)

).

Since the conservation of H1 implies that ‖u‖∞ stays bounded, therefore h2(t) → +∞implies that ux

2(t)→ +∞ as t→ T ∗. Moreover from (3.8)

ux′(t) ≤ u

2M2N2 +K1 < 0,

together with the fact that u′(t) > 0 we know that

uux(t, q)→ −∞, as t→ T ∗.

�

Remark 3.1. Using a similar argument one can prove the finite time blow-up for data

satisfying u0(x2) < 0 and

m0(x) < 0, x < x2; m0(x2) = 0; m0(x) > 0, x > x2,

u0(x2)

2

[u2

0(x2)− u20,x(x2)

]−K1 > 0.

3.2.2. Non-sign-changing momentum. Next we consider the Novikov equation with a weak

linear dispersion and also allow for a general initial momentum density m0. For a general

γ ∈ R, the sign-preservation of m does not hold. Moreover H2 as in (3.4) may not be

conserved. Thus H2 does not necessarily provide a bound for ‖ux‖4 directly. However, we

can find a substitute by slightly modifying H2. More precisely, we have


Proposition 3.1. Denote I(t) by

I(t) =

∫R

(u4 + 2u2u2

x −1

3u4x +

4

3γu2

)dx. (3.13)

Then there holds the following Morawetz-type identity

dI

dt= −4

3γ

∫Ruu3

xdx. (3.14)

Proof. Note that the Novikov equation (1.3) can be rewritten as

ut + u2ux + (1− ∂2x)−1∂x

(u3 +

3

2uu2

x + γu

)+

1

2(1− ∂2

x)−1u3x = 0. (3.15)

Differentiating (3.15) with respect to x leads to the expression for uxt

uxt =− u2uxx −1

2uu2

x + u3 + γu− (1− ∂2x)−1

(u3 +

3

2uu2

x + γu

)− 1

2(1− ∂2

x)−1∂xu3x.

A direct computation using (3.13) yields

dI

dt=

∫R

(4u3 + 4uu2

x + 4u2xuxx +

8

3γu

)utdx−

4

3

∫Ru3uxxt dx. (3.16)

Plugging

uxxt = ut + 4u2ux − 3uuxuxx − u2uxxx + γux

into (3.16), and using equation (3.15) to replace ut, the Morawetz-type identity (3.14) is

then obtained after integration by parts. �

We will see later that the quantity I(t) will play a similar role as H2 to control the

L4 norm of ux. Before we state the blow-up result, let us introduce some notations for

convenience. Let I0 = I(0) (I(t) is defined in (3.13)), and define

C1 =2√

2

3|γ|√H1[u0], C2 = −I0 +

4

3|γ|H1[u0] + H2

1 [u0],

C3 =3

2C1

√H1[u0], C4 =

√2H

321 [u0] +

(1

2|γ|+ 2

√3C2

)√H1[u0], and

K2(t) =√

2H321 [u0] +

(3C1t+ 2

√3C2

)√H1[u0].

(3.17)

Theorem 3.2. Let u0 ∈ Hs(R) for s > 52 . Suppose there exist some 0 < β < 1 and

x3 ∈ R such that u0(x3) > 0 and

u0,x(x3) < −B + 1

B − 1

√2K3(T2)

βu0(x3), (3.18)

where

B = exp(T2

√2βu0(x3)K3(T2)

),

T2 =−C4 +

√C2

4 + 4C3(1− β)u0(x3)

2C3, and


K3(t) =

(1 +

√2

2

)|γ|√H1[u0] +

(2− β)3

2u3

0(x3) +K2(t),

with C3, C4 and K2(t) defined in (3.17). Then the corresponding solution u(t, x) blows up

in finite time with an estimate of the blow-up time T ∗ as

T ∗ ≤ T3 :=1√

2βu0(x3)K3(T2)log

u0,x(x3)−√

2K3(T2)βu0(x3)

u0,x(x3) +√

2K3(T2)βu0(x3)

. (3.19)

Proof. Note that H1[u] is conserved and ‖u‖∞ ≤ 1√2‖u‖H1 . A simply computation shows

that ∫Ru2u2

xdx ≤ ‖u‖2∞∫Ru2xdx ≤ ‖u‖2∞‖u‖2H1 ≤

1

2‖u‖4H1 =

1

2H2

1 [u0] (3.20)

and ∫R

(u4 + 2u2u2

x +4

3γu2

)dx ≤ ‖u‖2∞‖u‖22 +

4

3|γ|‖u‖22 + 2

∫Ru2u2

xdx

≤ 4

3|γ|H1[u0] + H2

1 [u0].

(3.21)

We now use the quantity I(t) to derive some convolution estimates. Let f(t) =(∫R u

4xdx) 1

2 . By (3.14), (3.20) and the definition of C1, we get

−dIdt

=4

3γ

∫Ruu3

xdx ≤4

3|γ|(∫

Ru2u2

xdx

) 12(∫

Ru4x

) 12

≤ 2√

2

3|γ|√H1[u0]f(t) = C1f(t),

then

−I(t) ≤ C1

∫ t

0f(τ)dτ − I0.

According to the definition of I(t), we deduce

1

3f2(t) ≤ C1

∫ t

0f(τ)dτ − I0 +

∫R

(u4 + 2u2u2

x +4

3γu2

)dx.

This in turn implies that

f2(t) ≤ 3C1

∫ t

0f(τ)dτ + 3C2, (3.22)

where we have used (3.21) and the definition of C2. Denote the right-hand side of (3.22)

by g(t). It is then inferred from (3.22) that

g′(t) = 3C1f(t) ≤ 3C1g12 (t).

Integration over the interval [0, t] gives√g(t) ≤ 3

2C1t+

√g(0) =

3

2C1t+

√3C2,

so

f(t) ≤ √g ≤ 3

2C1t+

√3C2.


With the above inequality in hand, we can derive the following convolution estimates∣∣∣p± ∗ (u∓ ux)3∣∣∣ ≤ ‖p±‖∞ ∥∥∥(u∓ ux)3

∥∥∥1≤ 2

(‖u‖33 + ‖ux‖33

)≤ 2

√H31 [u0]

2+ ‖ux‖24

√H1[u0]

≤ 2

√H31 [u0]

2+

√H1[u0]

(3

2C1t+

√3C2

)= : K2(t).

(3.23)

We are now in the position to prove the blow-up result. In view of (3.6) and (3.23), it

follows that

|u′(t)| ≤ |γ||p ∗ ux|+1

2

∣∣p+ ∗ (u− ux)3 − p− ∗ (u+ ux)3∣∣

≤ |γ|‖p‖2‖ux‖2 +K2(t) ≤ 1

2|γ|‖u‖H1 +K2(t)

=1

2|γ|√H1[u0] +K2(t)

=: 2C3t+ C4,

(3.24)

where C3 and C4 are defined in (3.17). Integration over the time interval [0, t] yields

u0(x3)− C3t2 − C4t ≤ u(t) := u

(t, q(t, x3)

)≤ u0(x3) + C3t

2 + C4t, (3.25)

so

u(t) > 0 for 0 ≤ t < T1 :=−C4 +

√C2

4 + 4C3u0(x3)

2C3.

Applying the convolution estimate (3.23) to the dynamics of ux in (3.6), we have

ux′(t) ≤ u

2

(u2 − u2

x

)+ |γ||u− p ∗ u|+K2(t)

≤ u

2

(u2 − u2

x

)+ |γ| (‖u‖∞ + ‖p‖1‖u‖2) +K2(t)

≤ u

2

(u2 − u2

x

)+ |γ|

(√2

2+ 1

)√H1[u0] +K2(t).

(3.26)

We now consider inequality (3.25) on the time interval 0 ≤ t ≤ T2 =−C4+

√C2

4+4C3(1−β)u0(x3)

2C3.

It in turn implies that

0 < βu0(x3) ≤ u(t) ≤ (2− β)u0(x3).

Therefore, for 0 ≤ t ≤ T2, we deduce from (3.26) that

ux′(t) ≤ − β

2u0(x3)ux

2 +1

2(2− β)3u3

0(x3) + |γ|

(√2

2+ 1

)√H1[u0] +K2(t)

=: − β

2u0(x3)ux

2 +K3(t) ≤ −β2u0(x3)ux

2 +K3(T2).

(3.27)

It is observed from (3.18) that u0,x(x3) < −√

2K3(T2)βu0(x3) and T3 < T2. Then applying Lemma

3.3 to (3.27) implies ux(t) → −∞ as t → T ∗, where T ∗ is estimated in (3.19). Finally


notice that as t → T ∗, u(t) ≥ βu0(x3) > 0. The blow-up of ux thus implies the blow-up

of uux. This completes the proof of Theorem 3.2. �

In the same spirit we can establish the following result for the special case γ = 0.

Corollary 3.1. Let γ = 0, u0 ∈ Hs(R) for s > 5/2. Suppose there exist some 0 < δ < 1

and x3 ∈ R such that u0(x3) > 0 and

u0,x(x3) ≤ −A+ 1

A− 1

√(2− δ)3u3

0(x3) + 2K1

δu0(x3), (3.28)

where K1 is defined in (3.10) and

A = exp

(1− δ)u0(x3)√δu0(x3)

((2− δ)3u3

0(x3) + 2K1

)K1

.Then the corresponding solution u(t, x) blows up in finite time with an estimate of the


T ∗ ≤ T0 :=1√

δu0(x3)[(2− δ)3u30(x3) + 2K1]

log

u0,x(x3)−√

(2−δ)3u30(x3)+2K1

δu0(x3)

u0,x(x3) +√

(2−δ)3u30(x3)+2K1

δu0(x3)

.

(3.29)

Proof. Taking account of (3.7) and (3.12), we conclude that

u0(x3)−K1t ≤ u(t) := u(t, q(t, x3)) ≤ u0(x3) +K1t. (3.30)

This then implies that u(t) > 0 for 0 ≤ t < T+ := u0(x3)/K1. Using the convolution

estimates in (3.12) to the dynamics (3.8), we have

ux′(t) ≤ u

2(u2 − ux2) +K1.

Consider the bounds (3.30) on the time interval 0 ≤ t ≤ (1− δ)T+ we know that

0 < δu0(x3) ≤ u(t) ≤ (2− δ)u0(x3). (3.31)

Consequently,

ux′(t) ≤ −δ

2u0(x3)ux

2 +

((2− δ)3u3

0(x3)

2+K1

). (3.32)

Applying Lemma 3.3 to (3.32) we conclude that

ux(t)→ −∞ as t→ T ∗,

where T ∗ is estimated in (3.29), provided that

u0,x(x3) < −

√(2− δ)3u3

0(x3) + 2K1

δu0(x3), and T0 ≤ (1− δ)T+.

Solving the above we obtain the condition on the initial data as given in (3.28). Finally

notice that as t → T ∗, u(t) ≥ δu0(x3) > 0, hence the blow-up of ux implies the blow-up

of uux, which completes the proof of the corollary. �


4. Blow-up for the gmCH equation

In this section we focus on the finite-time blow-up of waves for the gmCH equation

(1.4). Similar to the proof of Theorem 6.2 in [27], we can establish the following blow-up

criterion for the gmCH equation.

Lemma 4.1. Suppose that u0 ∈ Hs(R) with s > 52 . Then the corresponding solution u to

the initial value problem (1.4) blows up in finite time T > 0 if and only if

limt→T

infx∈R

(k1m(t, x) + k2)ux(t, x) = −∞.

The characteristics associated to the gmCH equation (1.4) is determined as followsdq(t, x)

dt=(k1(u2 − u2

x) + k2u)

(t, q(t, x)),

q(0, x) = x,x ∈ R, t ∈ [0, T ). (4.1)

4.1. Dynamics along the characteristics. We now compute the dynamics of some

important quantities along the characteristic q(t, x) associated to the gmCH equation.

Denote ′ to be the derivative ∂t + [k1(u2 − u2x) + k2u]∂x along the characteristic.

Lemma 4.2. Let u0 ∈ Hs(R), s ≥ 3. Then u(t, q(t, x)), ux(t, q(t, x)) and m(t, q(t, x))

satisfy the following integro-differential equations

u′(t) =− 2

3k1u

3x − γpx ∗ u+

k1

3

[p+ ∗ (u− ux)3 − p− ∗ (u+ ux)3

]− k2px ∗

(u2 +

1

2u2x

), (4.2)

u′x(t) = k1

(1

3u3 − uu2

x

)+ k2

(u2 − 1

2u2x

)+ γ(u− p ∗ u)

− k1

3

[p+ ∗ (u− ux)3 + p− ∗ (u+ ux)3

]− k2p ∗

(u2 +

1

2u2x

), (4.3)

m′(t) =−(2k1m

2 + 2k2m+ γ)ux. (4.4)

Proof. In view of (1.4), we have

(1− ∂2x)[ut +

(k1(u2 − u2

x) + k2u)ux]

= k1

[−((u2 − u2

x)m)x

+ ux(u2 − u2x)−

(uxx(u2 − u2

x))x− 2

(u2xm)x

]+ k2[−2uxm− umx + uux − ∂x(u2

x + uuxx)]− γux= − k1

(2muux + 2(u2

xm)x)− k2(2uux + uxuxx)− γux,

which implies that

u′(t) = −γp ∗ ux − k1p ∗(2muux + 2

(u2xm)x

)− k2p ∗ (2uux + uxuxx). (4.5)

Plugging the identity (see (3.6) in [6])

2p ∗ (muux) + 2p ∗((u2xm)x

)= 2p ∗ (muux) + 2px ∗

(u2xm)

=2

3u3x −

1

3

(p+ ∗ (u− ux)3 − p− ∗ (u+ ux)3

)


into (4.5), we obtain (4.2). Next we calculate u′x. Differentiating (4.5) with respect to x,

we have

u′x = − γpx ∗ ux − k1

[(u2 − u2

x

)xux + px ∗

(2muux + 2

(u2xm)x

)]− k2

[u2x + px ∗ (2uux + uxuxx)

]= γ(u− p ∗ u)− 2k1

[px ∗ (muux) + p ∗

(u2xm)]

− k2

[1

2u2x − u2 + p ∗

(u2 +

1

2u2x

)].

(4.6)

Plugging the identity (see (3.7) in [6])

2px ∗ (muux) + 2p ∗(u2xm)

= uu2x −

1

3u3 +

1

3

[p+ ∗ (u− ux)3 + p− ∗ (u+ ux)3

],

we obtain (4.3). Finally our attention is turned to the estimate of m. On account of (1.4),

we deduce that

m′(t) = mt + [k1(u2 − u2x) + k2u]mx

= k1

[(u2 − u2

x)mx −((u2 − u2

x)m)x

]+ k2[umx − 2uxm− umx]− γux

= − 2k1uxm2 − 2k2uxm− γux,

thereby concluding the proof of Lemma 4.2. �

4.2. Non-sign-changing momentum. In this subsection, we derive some sufficient con-

ditions for the blow-up of the initial-value problem (1.4) when the parameter γ = 0. The

following lemma shows that, if m0 = (1 − ∂2x)u0 does not change sign, then m(t, x) will

not change sign for any t ∈ [0, T ). This conservative property of the momentum m will

be crucial in the proof of our blow-up result.

Lemma 4.3. Let u0 ∈ Hs(R), s > 52 , and let T > 0 be the maximal existence time

of the corresponding strong solution u to (1.4). Then (4.1) has a unique solution q ∈C1([0, T )× R,R) such that the map q(t, ·) is an increasing diffeomorphism of R with

qx(t, x) = exp

(∫ t

0

(2k1m+ k2

)ux(s, q(s, x))ds

)> 0, ∀ (t, x) ∈ [0, T )× R. (4.7)

Furthermore, for all (t, x) ∈ [0, T )× R it holds that

m(t, q(t, x)

)= m0(x) exp

(−2

∫ t

0(k1m+ k2)ux

(s, q(s, x)

)ds

). (4.8)

Proof. Since u ∈ C1([0, T ), Hs−1(R)

)and Hs(R) ↪→ C1(R), both u(t, x) and ux(t, x) are

bounded, Lipschitz in the space variable x, and of class C1 in time. Therefore, by the well-

known classical results in the theory of ordinary differential equations, the initial value

problem (4.1) has a unique solution q(t, x) ∈ C1 ([0, T )× R) .

Differentiating (4.1) with respect to x yieldsd

dtqx =

(2k1m+ k2

)ux(t, q)qx,

qx(0, x) = 1,x ∈ R, t ∈ [0, T ). (4.9)


The solution to (4.9) is given by (4.7). For every T ′ < T, it follows from the Sobolev

embedding that

sup(s,x)∈[0,T ′)×R

|(2k1m+ k2

)ux(s, x)| <∞.

It is inferred from (4.7) that there exists a constantK > 0 such that qx(t, x) ≥ e−Kt, (t, x) ∈[0, T )× R, which implies that the map q(t, ·) is an increasing diffeomorphism of R before

the blow-up time.

On the other hand, by (1.4) and (4.1), we have

d

dtm(t, q(t, x)

)= (mt +mxqt)

(t, q(t, x)

)= mt

(t, q(t, x)

)+[k1

(u2 − u2

x

)+ k2u

]mx

(t, q(t, x)

)= − 2 (k1mux + k2ux)m

(t, q(t, x)

).

Therefore, solving the equation with regard to m(t, q(t, x)

)leads to (4.8). This completes

the proof of Lemma 4.3. �

We now state the following result on the blow-up for a non-changing-sign momentum.

Theorem 4.1. Let k1 > 0, k2 ≥ 0, u0 ∈ Hs(R) for s > 52 and m0 ≥ 0. Suppose that

there exists a point x4 ∈ R such that

m0(x4) > 0 and u0,x(x4) < − 1√2

(u0(x4) +

3k2

2k1

). (4.10)



T ∗ ≤ − 1

2k1m0(x4)u0,x(x4).

Proof. As before, we will trace the dynamics along the characteristics emanating from x4.

Denote

u(t) = u (t, q(t, x4)) , ux(t) = ux (t, q(t, x4)) ,

m(t) = m (t, q(t, x4)) , M(t) = (mux) (t, q(t, x4)) .

Since we know that m0 ≥ 0, in particular, m0(x4) > 0, so from (4.8), we know that

m(t, x) ≥ 0 and m(t) > 0. Therefore from the identities

u(t, x) = p ∗m(t, x) =1

2

∫Re−|x−y|m(t, y)dy, ux(t, x) = px ∗m(t, x),

we have u(t, x) ≥ 0 and u(t) > 0. Moreover

u− ux = 2p+ ∗m ≥ 0, u+ ux = 2p− ∗m ≥ 0. (4.11)

Hence we know |ux(t, x)| ≤ u(t, x). Therefore ux does not blow up. From the blow-up

criterion in Lemma 4.1, it suffices to consider the quantity M = mux. Using (4.3) and

(4.4), a simple calculation then gives

M ′(t) = − 2k1M2 +

k1

3mu

(u2 − 3ux

2)

+ k2

(mu2 − 5

2mux

2

)− k1

3m[p+ ∗ (u− ux)3 + p− ∗ (u+ ux)3

]− k2mp ∗

(u2 +

1

2u2x

).


Taking account of (4.11) and the inequality p ∗(u2 + 1

2u2x

)≥ 1

2u2, it is then adduced that

M ′(t) ≤ −2k1M2 +

k1

3mu

(u2 − 3ux

2)

+k2

2m(u2 − 5ux

2).

Our argument is to find certain conditions on the initial data under which there holds

the Riccati-like inequality M ′(t) ≤ −CM2. Thus from the sign conditions u(t) > 0 and

m(t) > 0, we would like to have u2−3ux2 ≤ 0, and this would also imply that u2−5ux

2 ≤ 0.

That is to say, it suffices to recognize finite-time blow-up of M(t) if the ratio |ux/u| stays

big along the characteristics. This suggests us to trace the dynamics of ux/u. However,

due to the inhomogeneity of the nonlinearities, one can only show that(uxu

)′(t) ≤ C1

(u2 − 2ux

2 + C2u).

Thus a large (negative) ratio ux/u is not enough to make the right-hand side negative.

The way to resolve this is to absorb the linear term C2u into the quadratic one by replacing

u by a vertical shift u + a. Therefore instead, we will track the dynamics of ux/(u+ a)

along the characteristics, where a ≥ 0 will be chosen later.(uxu+ a

)′(t) =

k1

3(u+ a)2

(u2 − ux2

)(u2 − 2ux

2)

+k2

(u+ a)2

(u3 − 1

2uux

2

)+

ak1u

3(u+ a)2

(u2 − 3ux

2)

+ak2

2(u+ a)2

(2u2 − ux2

)− k1

3(u+ a)2(u+ ux + a) p+ ∗ (u− ux)3

(t, q(t, x4)

)− k1

3(u+ a)2(u− ux + a) p− ∗ (u+ ux)3

(t, q(t, x4)

)− k2

(u+ a)2(u+ ux + a) p+ ∗

(u2 +

1

2u2x

)(t, q(t, x4)

)− k2

(u+ a)2(u− ux + a) p− ∗

(u2 +

1

2u2x

)(t, q(t, x4)

).

Regrouping the terms and using the fact that p± ∗(u2 + 1

2u2x

)≥ 1

4u2 we have(

uxu+ a

)′(t) ≤ k1

3(u+ a)2

(u2 − ux2

)(u2 − 2ux

2)

+k2

2(u+ a)2

(u3 − uux2

)+

ak1u

3(u+ a)2

(u2 − 3ux

2)

+ak2

2(u+ a)2

(u2 − ux2

)=

k1

3(u+ a)2

[(u2 − ux2

)(u2 − 2ux

2 +3k2

2k1u+

3ak2

2k1

)+ au

(u2 − 3ux

2)]

≤ k1

3(u+ a)2

{(u2 − ux2

)[(u+

3k2

2k1

)2

− 2ux2 +

3k2

2k1

(a− 3k2

2k1

)]+au

(u2 − 3ux

2)}

.


So now choose a = 3k22k1≥ 0, then the above reads(

uxu+ a

)′(t) ≤ k1

3(u+ a)2

{(u2 − ux2

)[(u+

3k2

2k1

)2

− 2ux2

]+ au

(u2 − 3ux

2)}

.

By condition (4.10), we have chosen the initial data such that

ux(0) < − 1√2

(u(0) +

3k2

2k1

)≤ − 1√

2u(0) < − 1√

3u(0).

From which we see that the right-hand side of the above is negative initially. Hence

ux/

(u+

3k2

2k1

)decreases, and thus

ux

u+ 3k22k1

(t) <ux

u+ 3k22k1

(0) < − 1√2.

So u(t) + 3k22k1

+√

2ux(t) < 0. Therefore

u(t) +√

3ux(t) < 0 and u(t) +√

5ux(t) < 0. (4.12)

Meanwhile we also have

u(t)−√

3ux(t) > 0 and u(t)−√

5ux(t) > 0. (4.13)

Now plugging (4.12) and (4.13) into (4.2) we obtain

M ′(t) ≤ −2k1M2 +

k1

3mu

(u2 − 3ux

2)

+k2

2m(u2 − 5ux

2)≤ −2k1M

2,

and hence M(t) blows up in finite time with an estimate of the blow-up time T ∗ as

T ∗ ≤ − 1

2k1M(0)= − 1

2k1m0(x4)u0,x(x4),

This completes the proof of Theorem 4.1. �

4.3. Blow-up for a general momentum. In this subsection we turn our attention to

the general case where γ needs not equal to zero, and therefore there is no sign-preservation

for the momentum density as in the previous case.

We first derive some convolution estimates for later use. Our assumptions for the pa-

rameters throughout this subsection are k1 > 0, k2 ∈ R and γ ∈ R. From the conservation

laws (1.5) we infer that

‖ux‖44 = 3

∫R

(u4 + 2u2u2

x

)dx+

6k2

k1

∫R

(u3 + uu2

x

)dx

+6γ

k1

∫Ru2dx− 3

k1H2[u0]

≤(

4‖u‖2∞ +

∣∣∣∣6k2

k1

∣∣∣∣ ‖u‖∞ +

∣∣∣∣6γk1

∣∣∣∣)H1[u0]− 3

k1H2[u0]

≤(

2H1[u0] + 3√

2

∣∣∣∣k2

k1

∣∣∣∣H 121 [u0] +

∣∣∣∣6γk1

∣∣∣∣)H1[u0]− 3

k1H2[u0]

(4.14)


and this in turn gives the convolution estimates

|p± ∗ (u∓ ux)3 | ≤ ‖p±‖∞‖ (u∓ ux)3 ‖1 ≤ 2(‖u‖33 + ‖ux‖33

)≤ 2

[(2H1[u0] + 3

√2

∣∣∣∣k2

k1

∣∣∣∣H 121 [u0] +

∣∣∣∣6γk1

∣∣∣∣)H21 [u0]− 3

k1H1[u0]H2[u0]

] 12

+√

2H321 [u0] =: Q.

(4.15)

We can further utilize the conservation of H1[u0] to bound

|px ∗ u| = |(p− − p+) ∗ u| ≤ (‖p−‖1 + ‖p+‖1)‖u‖∞

≤ 1√2‖u‖H1 =

√H1[u0]

2,

|u− p ∗ u| ≤ 2‖u‖∞ ≤2√2‖u‖H1 =

√2H1[u0].∣∣∣∣p ∗ (u2 +

1

2u2x

)∣∣∣∣ ≤ ‖p‖∞ ∥∥∥∥u2 +1

2u2x

∥∥∥∥1

≤ 1

2H1[u0].

(4.16)

Our blow-up theorem for the gmCH equation (1.4) with general γ can be stated as follows.

Theorem 4.2. Let γ ∈ R, k1 > 0 and u0 ∈ Hs(R) with s > 52 . Assume that there exists

an x5 ∈ R such that

u0(x5) > max

{0,− k2

2k1

}, m0(x5) > max{α, 0},

u0,x(x5) ≤ min

{−(A1)

13 ,−

√A2

k1u0(x5) + k22

},

(4.17)

where

A1 = Q+3|γ|2k1

√H1[u0]

2+

3|k2|4k1

H1[u0],

A2 =2

3k1Q+ |γ|

√2H1[u0] + |k2|H1[u0] +

k1

6√

2H

321 [u0],

with Q defined in (4.15), and

α =

−k2 +

√∆

2k1, if ∆ = (k2

2 − 2k1γ) > 0,

− k2

2k1, if ∆ = (k2

2 − 2k1γ) ≤ 0.


Then the solution u(t, x) blows up in finite time with an estimate of the blow-up time T ∗

as

T ∗ ≤

− 1

2u0,x(x5)√

∆log

m0(x5) + k2+√

∆2k1

m0(x5) + k2−√

∆2k1

, if ∆ > 0,

− 1

2k1u0,x(x5)(m0(x5) + k2

2k1

) , if ∆ = 0,

−

π

2− arctan

2k1m0(x5) + k2√−∆

u0,x(x5)√−∆

, if ∆ < 0.

(4.18)

Proof. Plugging the estimates (4.15)-(4.16) into (4.2) and (4.3), we deduce that

u′(t) = u′ (t, q(t, x5)) ≥ −2

3k1ux

3 − |γ|√H1[u0]

2− 2

3k1Q−

1

2|k2|H1[u0],

ux′(t) = u′x (t, q(t, x5)) ≤ −

(k1u+

1

2k2

)ux

2 +2

3k1Q+ |γ|

√2H1[u0]

+ |k2|H1[u0] +k1

6√

2H

321 [u0].

So we know that u(t) is increasing when

ux3(t) ≤ −

(Q+

3|γ|2k1

√H1[u0]

2+

3|k2|4k1

H1[u0]

)=: −A1

and ux(t) is decreasing when(k1u+

1

2k2

)ux

2 ≥ 2

3k1Q+ |γ|

√2H1[u0] + |k2|H1[u0] +

k1

6√

2H

321 [u0] =: A2.

Hence from the assumption on the initial data (4.17), we know that over the time of

existence of solutions u(t) is increasing and ux(t) is decreasing. In particular,

u(t) ≥ u0(x5) > 0, ux(t) ≤ u0,x(x5) < 0. (4.19)

By the assumption (4.17) on m0(x5), we know that

2k1m20(x5) + 2k2m0(x5) + γ > 0.

Using (4.19), we deduce from (4.4) that over the time of solutions m(t) is increasing, so

m(t) > m0(x5) and

m′(t) := m′ (t, q(t, x5)) = −ux(t)(2k1m

2(t) + 2k2m(t) + γ)

≥ − u0,x(x5)(2k1m

2(t) + 2k2m(t) + γ)> 0.

(4.20)

We show that blow-up must occur in finite time. The proof is divided into three cases.

Case 1: ∆ > 0. Let y1 and y2 be the two distinct real roots of the equation 2k1y2 +

2k2y + γ = 0, so

y1,2 =−k2 ∓

√∆

2k1, y1 < y2,

2k1y2 + 2k2y + γ = 2k1(y − y1)(y − y2).


Integrating inequality (4.20) over [0, t] gives

m(t) ≥ y2 − y1E(t)

1− E(t)→ +∞, as t→ − 1

2k1u0,x(x5)(y2 − y1)log

(m0(x5)− y1

m0(x5)− y2

),

where

E(t) =m0(x5)− y2

m0(x5)− y1exp

(− 2k1u0,x(x5)(y2 − y1)t

).

Notice that ux(t) ≤ u0,x(x5) < 0. It is then deduced that in this case

(k1m+ k2)ux(t, q(t, x5)

)→ −∞, as t→ − 1

2k1u0,x(x5)(y2 − y1)log

(m0(x5)− y1

m0(x5)− y2

).

Case 2: ∆ = 0. In this case the equation 2k1y2 + 2k2y + γ = 0 has the unique real root

y = − k22k1

, and so

2k1y2 + 2k2y + γ = 2k1

(y +

k2

2k1

)2

.

Integrating inequality (4.20) over the interval [0, t] gives

m(t) ≥m0(x5) + k2

2k1

2k1u0,x(x5)(m0(x5) + k2

2k1

)t+ 1

− k2

2k1→ +∞,

as t→ − 1

2k1u0,x(x5)(m0(x5) + k2

2k1

) ,which, similar to Case 1, implies that

(k1m+ k2)ux(t, q(t, x5)

)→ −∞, as t→ − 1

2k1u0,x(x5)(m0(x5) + k2

2k1

) .Case 3: ∆ < 0. In this case we have

2k1y2 + 2k2y + γ = 2k1

(m+

k2

2k1

)2

− ∆

2k1.

Integrating inequality (4.20) over the interval [0, t] gives

m(t) ≥ − k2

2k1+

√−∆

2k1tan

−u0,x(x5)√−∆t+ arctan

2k1

(m0(x5) + k2

2k1

)√−∆

,which goes to +∞ as

t→

π

2− arctan

2k1m0(x5) + k2√−∆

−u0,x(x5)√−∆

.

Hence, we deduce that

(k1m+ k2)ux(t, q(t, x5)

)→ −∞, as t→

π

2− arctan

2k1m0(x5) + k2√−∆

−u0,x(x5)√−∆

.

This completes the proof of Theorem 4.2. �

Following the proof of Theorem 4.2, we can deal with the case of γ = 0 and k2 < 0.

Indeed, we have the following result, which can not be obtained from Theorem 4.1 .


Corollary 4.1. Let k1 > 0, k2 < 0, γ = 0 and u0 ∈ Hs(R) with s > 52 and m0(x) ≥

0, ∀x ∈ R. Assume that there exists a point x6 ∈ R such that

u0(x6) > − k2

2k1, m0(x6) > −k2

k1,

u0,x(x6) ≤ min

{− (A3)

13 ,−

√A4

k1u0(x6) + k22

},

where

A3 =Q

2− 3k2

4k1H1[u0], A4 =

k1

6√

2H

3/21 [u0]− k2

2H1[u0],

and Q is defined by (4.15). Then the solution u(t, x) blows up in finite time with an

estimate of the blow-up time T ∗ as

T ∗ ≤ 1

2k2u0,x(x6)log

m0(x6)

m0(x6) + k2k1

.

Acknowledgments. The work of Chen is partially supported by the Central Research

Development Fund-04.13205.30205 from University of Pittsburgh. The work of Guo is

partially supported by the NSFC grant-11271192 and the Priority Academic Program

Development of Jiangsu Higher Education Institution. The work of Liu is partially sup-

ported by the NSF grant DMS-1207840 and the NSFC grant-11271192. The work of Qu is

supported in part by the NSFC grant-11471174 and NSF of Ningbo grant-2014A610018.

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(Robin Ming Chen) Department of Mathematics, University of Pittsburgh, PA 15260, USA

E-mail address: [email protected]

(Fei Guo) School of Mathematical Sciences and Jiangsu Key Laboratory for NSLSCS,

Nanjing Normal University, Nanjing 210023, P.R. China


(Yue Liu) Department of Mathematics, University of Texas at Arlington, Arlington, TX

76019, USA; Ningbo University, Ningbo 315211, China


(Changzheng Qu) Department of Mathematics, Ningbo University, Ningbo 315211, P. R.

China


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